A way is described to find the initial conditions
for the simplest three-dimensional periodic motions (3DPMs)
in the vicinity of the equilibrium points (EPs)
of an elongated asteroid whose shape is approximated
by a triaxial ellipsoid (TE).
A condition for the existence of these 3DPMs is formulated
which may or may not be satisfied,
depending on the spin period of the asteroid.
A closed integral form of the gravitational potential
of a TE is used.
The general formulae for the computation of the initial conditions
and some parameters of a 3DPM are derived
from analysing the variational equations of motion.
These initial conditions are used to integrate numerically
the non-linear equations of motion in the vicinity
of the EPs of a specific model of a triaxial ellipsoid
based on the asteroid 243 Ida for a number of different
rotation periods.
We found that the period and some parameters of a 3DPM
obtained by the numerical integration (with initial conditions
predicted from the variational equations) have approximately
the values computed from the variational equations. We show that the equatorial projections of all 3DPMs
constructed in the current work are ellipses
with sizes dependent on one free parameter.
General formulae are presented for the computation of the ratio
of the semi-axes of the ellipse and the period of motion in it. A special study is made for a boundary case where the EPs
lying along the greatest ellipsoid axis (strongly unstable)
are touching the asteroid surface. This critical value
of the spin period of the asteroid (at which the loss of loose regoliths
or destruction of the asteroid can start) is shown to be the same
for the asteroids with similar densities and axial ratios. Some comments are given on the condition under which a 3-D periodic
motion can exist in the vicinity of the EPs of the Jacobi ellipsoid.

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.